References

↑ abThese criteria assume that all voters vote their true preference order. This is problematic for Approval and Range, where various votes are consistent with the same order. See approval voting for compliance under various voter models.

↑ abcdeIn Approval, Range, and Majority Judgment, if all voters have perfect information about each other's true preferences and use rational strategy, any Majority Condorcet or Majority winner will be strategically forced – that is, win in the unique Strong Nash equilibrium. In particular if every voter knows that "A or B are the two most-likely to win" and places their "approval threshold" between the two, then the Condorcet winner, if one exists and is in the set {A,B}, will always win. These systems also satisfy the majority criterion in the weaker sense that any majority can force their candidate to win, if it so desires. (However, as the Condorcet criterion is incompatible with the participation criterion and the consistency criterion, these systems cannot satisfy these criteria in this Nash-equilibrium sense. Laslier, J.-F. (2006) "Strategic approval voting in a large electorate,"IDEP Working Papers No. 405 (Marseille, France: Institut D'Economie Publique).)

↑ abThe original independence of clones criterion applied only to ranked voting methods. (T. Nicolaus Tideman, "Independence of clones as a criterion for voting rules", Social Choice and Welfare Vol. 4, No. 3 (1987), pp. 185–206.) There is some disagreement about how to extend it to unranked methods, and this disagreement affects whether approval and range voting are considered independent of clones. If the definition of "clones" is that "every voter scores them within ±ε in the limit ε→0+", then range voting is immune to clones.

↑ abApproval and Plurality do not allow later preferences. Technically speaking, this means that they pass the technical definition of the LNH criteria - if later preferences or ratings are impossible, then such preferences can not help or harm. However, from the perspective of a voter, these systems do not pass these criteria. Approval, in particular, encourages the voter to give the same ballot rating to a candidate who, in another voting system, would get a later rating or ranking. Thus, for approval, the practically meaningful criterion would be not "later-no-harm" but "same-no-harm" - something neither approval nor any other system satisfies.

↑The number of piles that can be summed from various precincts is floor((e-1) N!) - 1.

↑Each prospective Kemeny-Young ordering has score equal to the sum of the pairwise entries that agree with it, and so the best ordering can be found using the pairwise matrix.

↑Bucklin voting, with skipped and equal-rankings allowed, meets the same criteria as Majority Judgment; in fact, Majority Judgment may be considered a form of Bucklin voting. Without allowing equal rankings, Bucklin's criteria compliance is worse; in particular, it fails Independence of Irrelevant Alternatives, which for a ranked method like this variant is incompatible with the Majority Criterion.

↑Majority judgment passes the rated majority criterion (a candidate rated solo-top by a majority must win). It does not pass the ranked majority criterion, which is incompatible with Independence of Irrelevant Alternatives.

↑Majority judgment passes the "majority condorcet loser" criterion; that is, a candidate who loses to all others by a majority cannot win. However, if some of the losses are not by a majority (including equal-rankings), the Condorcet loser can, theoretically, win in MJ, although such scenarios are rare.

↑Balinski and Laraki, Majority Judgment's inventors, point out that it meets a weaker criterion they call "grade consistency": if two electorates give the same rating for a candidate, then so will the combined electorate. Majority Judgment explicitly requires that ratings be expressed in a "common language", that is, that each rating have an absolute meaning. They claim that this is what makes "grade consistency" significant. MJ. Balinski M. and R. Laraki (2007) «A theory of measuring, electing and ranking». Proceedings of the National Academy of Sciences USA, vol. 104, no. 21, 8720-8725.

↑Majority judgment can actually pass or fail reversal symmetry depending on the rounding method used to find the median when there are even numbers of voters. For instance, in a two-candidate, two-voter race, if the ratings are converted to numbers and the two central ratings are averaged, then MJ meets reversal symmetry; but if the lower one is taken, it does not, because a candidate with ["fair","fair"] would beat a candidate with ["good","poor"] with or without reversal. However, for rounding methods which do not meet reversal symmetry, the chances of breaking it are on the order of the inverse of the number of voters; this is comparable with the probability of an exact tie in a two-candidate race, and when there's a tie, any method can break reversal symmetry.

↑Majority Judgment is summable at order KN, where K, the number of ranking categories, is set beforehand.

↑Range satisfies the mathematical definition of IIA, that is, if each voter scores each candidate independently of which other candidates are in the race. However, since a given range score has no agreed-upon meaning, it is thought that most voters would either "normalize" or exaggerate their vote such that it votes at least one candidate each at the top and bottom possible ratings. In this case, Range would not be independent of irrelevant alternatives. Balinski M. and R. Laraki (2007) «A theory of measuring, electing and ranking». Proceedings of the National Academy of Sciences USA, vol. 104, no. 21, 8720-8725.

Delegated votes are considered to be equivalent to voting the candidate's predeclared preferences.

Ballots only are considered (In other words, voters are assumed not to have preferences that cannot be expressed by a delegated or approval vote.)

Since at the time of assigning approvals on delegated votes there is always enough information to find an optimum strategy, candidates are assumed to use such a strategy.

↑For up to 4 candidates, SODA is monotonic. For more than 4 candidates, it is monotonic for adding an approval, for changing from an approval to a delegation ballot, and for changes in a candidate's preferences. However, if changes in a voter's preferences are executed as changes from a delegation to an approval ballot, such changes are not necessarily monotonic with more than 4 candidates.

↑ abcFor up to 4 candidates, SODA meets the Consistency, Participation, IIA, and Cloneproof criteria. It can fail these criteria in certain rare cases with more than 4 candidates. This is considered here as a qualified success for the Consistency and Participation criteria, which do not intrinsically have to do with numerous candidates, and as a qualified failure for the IIA and Cloneproof criteria, which do.

↑SODA voting passes reversal symmetry for all scenarios that are reversible under SODA; that is, if each delegated ballot has a unique last choice. In other situations, it is not clear what it would mean to reverse the ballots, but there is always some possible interpretation under which SODA would pass the criterion.

↑SODA voting is always polytime computable. There are some cases where the optimal strategy for a candidate assigning delegated votes may not be polytime computable; however, such cases are entirely implausible for a real-world election.

↑Later preferences are only possible through delegation, that is, if they agree with the predeclared preferences of the favorite.

↑Random winner: Uniformly randomly chosen candidate is winner. Arbitrary winner: some external entity, not a voter, chooses the winner. These systems are not, properly speaking, voting systems at all, but are included to show that even a horrible system can still pass some of the criteria.

↑Random ballot: Uniformly random-chosen ballot determines winner. This and closely related systems are of mathematical interest because they are the only possible systems which are truly strategy-free, that is, your best vote will never depend on anything about the other voters. They also satisfy both consistency and IIA, which is impossible for a deterministic ranked system. However, this system is not generally considered as a serious proposal for a practical method.